Conics
Chords subtending right angle at origin
MJAT_TS3_P2
Grade 12
Question:
All chords of the curve $3x^2-y^2-2x+4y=0$ which subtend a right angle at the origin pass through:
A) Centre of the rectangular hyperbola $x^2-y^2-2x-4y=12$
B) The point of intersection of the lines $y+2x=0$ and $x=1$
C) The vertex of the parabola $x^2-2x-4y-7=0$
D) Centre of the circle $x^2+y^2+2x-4y-4=0$
Step-by-Step Solution
Key Concept: Homogenise $3x^2-y^2-2x+4y=0$ with the line $y=mx+c$ (i.e., $(2x-4y)/c=(mx/c-1)$... no: use $1=(2x-4y+\ldots)/c$). The condition for right angles at origin: sum of coefficients of $x^2$ and $y^2$ in the homogenised equation $=0$. This gives $m=-(c+2)$, so the chord $y+2x+c(x-1)=0$ passes through $(1,-2)$ for all $c$.
Fixed point $(1,-2)$. A ✓, B ✓, C ✓, D ✗. Answer: A, B, C.
Correct Answer: ABC